The cohomology approximation conjecture for random cubical covers

Let $T$ be a pure $(k+1)$-dimensional cubical complex. Let $T^\Box_{h,d}$ denote the uniform measure on $(k+1)$-dimensional, $d$-regular cubical complexes equipped with a cubical map $\tau$ to $T$ that is generically $hd$ to $1$.

Cohomology approximation conjecture. If $d \geq 4(2k+1)^2$, then, as $h$ tends to infinity, the induced map $\tau^*$ on the $k$th cohomology groups is an isomorphism to $\widetilde{H}^k(T)$.

This conjecture extends the preceding examples involving maps to a cube or a torus to arbitrary pure cubical complexes. The supplied text gives no resolution or further conditions, so the conjectural asymptotic assertion remains open.